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We present the &mu; -calculus, a syntax for &lambda;-calculus + control operators exhibiting symmetries such as program/context and call-by-name/call-by-value. This calculus is derived from implicational Gentzen's sequent calculus <i>LK</i>, a key classical logical system in proof theory. Under the Curry-Howard correspondence between proofs and… (More)
The λ&sgr;-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λ&sgr;-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations.
We review the relationship between abstract machines for (call-by-name or call-by-value) λ-calculi (extended with Felleisen's C) and sequent calculus , reintroducing on the way Curien-Herbelin's syntactic kit of the duality of computation. We provide a term language for a presentation of LK (with conjunction , disjunction, and negation), and we transcribe… (More)
We consider the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context. Our thesis is that the… (More)
A polymorphic function is parametric if its behavior does notdepend on the type at which it is instantiated. Starting with Reynolds'work, the study of parametricity is typically semantic. In this paper,we develop a syntactic approach to parametricity, and a formal systemthat embodies this approach: system… (More)
Categorical combinators [Curien 1986/1993; Hardin 1989; Yokouchi 1989] and more recently λ&sgr;-calculus [Abadi 1991; Hardin and Le´vy 1989], have been introduced to provide an explicit treatment of substitutions in the λ-calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special… (More)